I'm not sure what the attraction in playing youth footie for Chelsea or City is other than the facilities, seems a dead end from the progression into the first team stand point
Chelsea were bringing Izzy Brown (now at Brentford or something) & Solanke off the bench last season and Ihenancho (however the fuck it's spelt) is the only other I recall from the City youth team making an impact in the first team
If you your son was 10-13, a decent player for his age group and had an offer from City/Chelsea, you'd have a number of options
1. Take the offer.
They'll set your child up in one of the top private schools in the country and guarantee education until after GCSEs (even if your child drops out). The wage and benefits that the child and your family will get will likely far surpass anything you can provide for your family. In essence, a contract until their late teens would set your child up for the majority of their life, even as a player in their youth set up that has little chance of breaking through to the first team. They'll have access to some of the best coaches and facilities in the world.
2. Don't take the offer.
Depending on the club, the salary offered could be <£10k per year (e.g ours), or slightly more. Your son will have a higher chance of breaking through to the first team, and in all likelihood greater focus on their progressoin as an individual. The salary offered will not set them up for life. Statistically the chance of them making it and playing in one of the top divisions in Europe is still MINIMAL. They are 1 injury away from dropping out of the system completely, and even if they do remain health, there are no guarantees that they will make it as a pro. Should they drop out, it's likely that due to the focus on football during their youth they may have given up on opportunities for many other "high flying" careers (STEM, etc), compounded with the heart-break of falling out of the system in the desired career they have spent 10+ years grafting at.
Given the small possibility of breaking through to the first team at any club (unless you are one of the best in the world at your age, and even then...) What would you choose?